Asymptotic properties of parametric and nonparametric probability density estimators of sample maximum

TL;DR

This paper analyzes the asymptotic behavior of parametric and nonparametric estimators for the probability density of sample maxima, highlighting the advantages of nonparametric methods especially when the extreme value index is near zero.

Contribution

It introduces two nonparametric density estimators for sample maxima that outperform parametric methods under certain conditions, with theoretical and numerical validation.

Findings

Nonparametric estimators have faster convergence rates than parametric ones when gamma > -1.

Plug-in kernel estimator performs better than block-maxima-based estimator for gamma near zero.

Numerical results favor nonparametric estimators over parametric fitting as sample size increases.

Abstract

Asymptotic properties of three estimators of probability density function of sample maximum $f_{(m)}:=mfF^{m-1}$ are derived, where $m$ is a function of sample size $n$. One of the estimators is the parametrically fitted by the approximating generalized extreme value density function. However, the parametric fitting is misspecified in finite $m$ cases. The misspecification comes from mainly the following two: the difference $m$ and the selected block size $k$, and the poor approximation $f_{(m)}$ to the generalized extreme value density which depends on the magnitude of $m$ and the extreme index $\gamma$. The convergence rate of the approximation gets slower as $\gamma$ tends to zero. As alternatives two nonparametric density estimators are proposed which are free from the misspecification. The first is a plug-in type of kernel density estimator and the second is a block-maxima-based kernel density estimator. Theoretical study clarifies the asymptotic convergence rate of the plug-in type estimator is faster than the block-maxima-based estimator when $\gamma> -1$. A numerical comparative study on the bandwidth selection shows the performances of a plug-in approach and cross-validation approach depend on $\gamma$ and are totally comparable. Numerical study demonstrates that the plug-in nonparametric estimator with the estimated bandwidth by either approach overtakes the parametrically fitting estimator especially for distributions with $\gamma$ close to zero as $m$ gets large.